@article{Naden:anisotropic,
author = {Naden, Emma and M\"{a}rz, Thomas and Macdonald, Colin B.},
title = {Anisotropic diffusion on curved surfaces},
note = {Undergoing revision},
year = {2014},
url = {http://people.maths.ox.ac.uk/macdonald/AnisoDiffusionSurf.pdf},
archiveprefix = {arXiv},
eprint = {1403.2131},
optprimaryclass = {math.NA},
abstract = {We demonstrate a method for filtering images defined on
curved surfaces embedded in 3D. Applications are
noise removal and the creation of artistic effects.
Our approach relies on in-surface diffusion: we
formulate Weickert's edge/coherence enhancing
diffusion models in a surface-intrinsic way. These
diffusion processes are anisotropic and the
equations depend non-linearly on the data. The
surface-intrinsic equations are dealt with the
closest point method, a technique for solving
partial differential equations (PDEs) on general
surfaces. The resulting algorithm has a very simple
structure: we merely alternate a time step of a 3D
analog of the in-surface PDE in a narrow 3D band
containing the surface with a reconstruction of the
surface function. Surfaces are represented by a
closest point function. This representation is
flexible and the method can treat very general
surfaces. Experimental results include image
filtering on smooth surfaces, open surfaces, and
general triangulated surfaces.}
}
@article{vonGlehnMarzMacdonald:cpmol,
author = {von Glehn, Ingrid and M\"{a}rz, Thomas and Macdonald, Colin B.},
title = {An embedded method-of-lines approach to solving partial
differential equations on surfaces},
note = {Undergoing revision},
url = {http://people.maths.ox.ac.uk/macdonald/vonGlehnMarzMacdonald-2013-cpmol.pdf},
year = {2014},
archiveprefix = {arXiv},
eprint = {1307.5657},
optprimaryclass = {math.NA},
abstract = {We introduce a method-of-lines formulation of the
closest point method, a numerical technique for
solving partial differential equations (PDEs)
defined on surfaces. This is an embedding method,
which uses an implicit representation of the surface
in a band containing the surface. We define a
modified equation in the band, obtained in a
straightforward way from the original evolution PDE,
and show that the solutions of this equation are
consistent with those of the surface equation. The
resulting system can then be solved with standard
implicit or explicit time-stepping schemes, and the
solutions in the band can be restricted to the
surface. Our derivation generalizes existing
formulations of the closest point method and is
amenable to standard convergence analysis.}
}
@article{adaptive-RIDC,
author = {Christlieb, Andrew J. and Macdonald, Colin B. and Ong, Benjamin W. and Spiteri, Raymond J.},
title = {Revisionist Integral Deferred Correction with Adaptive Stepsize Control},
journal = {Comm. App. Math. Com. Sc.},
fjournal = {Communications in Applied Mathematics and Computational Science},
note = {To appear},
year = {2015},
tododoi = {10.2140/camcos.2015 TODO},
url = {http://people.maths.ox.ac.uk/macdonald/adaptive-ridc.pdf},
archiveprefix = {arXiv},
eprint = {1310.6331},
optprimaryclass = {math.NA},
abstract = {Adaptive stepsize control is a critical feature for the
robust and efficient numerical solution of
initial-value problems in ordinary differential
equations. In this paper, we show that adaptive
stepsize control can be incorporated within a family
of parallel time integrators known as Revisionist
Integral Deferred Correction (RIDC) methods. The
RIDC framework allows for various strategies to
implement stepsize control, and we report results
from exploring a few of them.}
}
@article{ChenMacdonald:ellipticCPM,
author = {Chen, Yujia and Macdonald, Colin B.},
title = {The {C}losest {P}oint {M}ethod and multigrid solvers for elliptic
equations on surfaces},
journal = {SIAM J. Sci. Comput.},
fjournal = {SIAM Journal on Scientific Computing},
year = {2015},
volume = {37},
number = {1},
optpages = {A134--A155},
doi = {10.1137/130929497},
url = {http://people.maths.ox.ac.uk/macdonald/ChenMacdonald-2015-elliptic-siam.pdf},
archiveprefix = {arXiv},
eprint = {1307.4354},
optprimaryclass = {math.NA},
abstract = {Elliptic partial differential equations are important
both from application and analysis points of
views. In this paper we apply the Closest Point
Method to solving elliptic equations on general
curved surfaces. Based on the closest point
representation of the underlying surface, we
formulate an embedding equation for the surface
elliptic problem, then discretize it using standard
finite differences and interpolation schemes on
banded, but uniform Cartesian grids. We prove the
convergence of the difference scheme for the
Poisson's equation on a smooth closed curve. In
order to solve the resulting large sparse linear
systems, we propose a specific geometric multigrid
method in the setting of the Closest Point
Method. Convergence studies both in the accuracy of
the difference scheme and the speed of the multigrid
algorithm show that our approaches are effective.}
}
@inproceedings{BiddleGlehnMacdonaldMarz:denoising,
author = {Biddle, Harry and von Glehn, Ingrid and Macdonald, Colin B. and M\"arz, Thomas},
title = {A volume-based method for denoising on curved surfaces},
booktitle = {Proc. ICIP13, 20th {IEEE} International Conference on Image Processing},
year = {2013},
pages = {529--533},
doi = {10.1109/ICIP.2013.6738109},
url = {http://people.maths.ox.ac.uk/macdonald/pmsurf.pdf},
abstract = {We demonstrate a method for removing noise from images
or other data on curved surfaces. Our approach
relies on in-surface diffusion: we formulate both
the Gaussian diffusion and Perona--Malik
edge-preserving diffusion equations in a
surface-intrinsic way. Using the Closest Point
Method, a recent technique for solving partial
differential equations (PDEs) on general surfaces,
we obtain a very simple algorithm where we merely
alternate a time step of the usual Gaussian
diffusion (and similarly Perona--Malik) in a small
3D volume containing the surface with an
interpolation step. The method uses a closest point
function to represent the underlying surface and can
treat very general surfaces. Experimental results
include image filtering on smooth surfaces, open
surfaces, and general triangulated surfaces.}
}
@article{KetchesonMacdonaldRuuth:SPERK,
author = {Ketcheson, David I. and Macdonald, Colin B. and Ruuth, Steven J.},
title = {Spatially Partitioned Embedded {R}unge--{K}utta Methods},
year = {2013},
fjournal = {SIAM Journal on Numerical Analysis},
journal = {SIAM J. Numer. Anal.},
url = {http://people.maths.ox.ac.uk/macdonald/sperk-siam.pdf},
volume = {51},
number = {5},
pages = {2887--2910},
archiveprefix = {arXiv},
eprint = {1301.4006},
optprimaryclass = {math.NA},
abstract = {We study spatially partitioned embedded Runge--Kutta
(SPERK) schemes for partial differential equations
(PDEs), in which each of the component schemes is
applied over a different part of the spatial domain.
Such methods may be convenient for problems in which
the smoothness of the solution or the magnitudes of
the PDE coefficients vary strongly in space. We
focus on embedded partitioned methods as they offer
greater efficiency and avoid the order reduction
that may occur in non-embedded schemes. We
demonstrate that the lack of conservation in
partitioned schemes can lead to non-physical effects
and propose conservative additive schemes based on
partitioning the fluxes rather than the ordinary
differential equations. A variety of SPERK schemes
are presented, including an embedded pair suitable
for the time evolution of fifth-order weighted
non-oscillatory (WENO) spatial discretizations.
Numerical experiments are provided to support the
theory.}
}
@article{MacdonaldMerrimanRuuth:ptclouds,
author = {Macdonald, Colin B. and Merriman, Barry and Ruuth, Steven J.},
title = {Simple computation of reaction-diffusion processes on point clouds},
year = {2013},
journal = {Proc. Natl. Acad. Sci.},
fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
doi = {10.1073/pnas.1221408110},
volume = {110},
number = {23},
opturl = {http://people.maths.ox.ac.uk/macdonald/TODO_after_embargo.pdf},
abstract = {The study of reaction-diffusion processes is much more
complicated on general curved surfaces than on
standard Cartesian coordinate spaces. Here we show
how to formulate and solve systems of
reaction-diffusion equations on surfaces in an
extremely simple way, using only the standard
Cartesian form of differential operators, and a
discrete unorganized point set to represent the
surface. Our method decouples surface geometry from
the underlying differential operators. As a
consequence, it becomes possible to formulate and
solve rather general reaction-diffusion equations on
general surfaces without having to consider the
complexities of differential geometry or
sophisticated numerical analysis. To illustrate the
generality of the method, computations for surface
diffusion, pattern formation, excitable media, and
bulk-surface coupling are provided for a variety of
complex point cloud surfaces.}
}
@article{Hadjimichael:efssp,
author = {Hadjimichael, Yiannis and Macdonald, Colin B. and
Ketcheson, David I. and Verner, James H.},
title = {Strong stability preserving explicit {R}unge--{K}utta
methods of maximal effective order},
year = {2013},
fjournal = {SIAM Journal on Numerical Analysis},
journal = {SIAM J. Numer. Anal.},
volume = {51},
number = {4},
pages = {2149--2165},
doi = {10.1137/120884201},
url = {http://people.maths.ox.ac.uk/macdonald/Hadjimichael-efssp-sinum.pdf},
archiveprefix = {arXiv},
eprint = {1207.2902},
optprimaryclass = {math.NA},
abstract = {We apply the concept of effective order to strong
stability preserving (SSP) explicit Runge--Kutta
methods. Relative to classical Runge--Kutta
methods, methods with an effective order of accuracy
are designed to satisfy a relaxed set of order
conditions, but yield higher order accuracy when
composed with special starting and stopping methods.
We show that this allows the construction of
four-stage SSP methods with effective order four
(such methods cannot have classical order four).
However, we also prove that effective order five
methods---like classical order five
methods---require the use of non-positive weights
and so cannot be SSP. By numerical optimization, we
construct explicit SSP Runge--Kutta methods up to
effective order four and establish the optimality of
many of them. Numerical experiments demonstrate the
validity of these methods in practice.}
}
@article{Maerz/Macdonald:cpfunctions,
author = {M\"{a}rz, Thomas and Macdonald, Colin B.},
title = {Calculus on surfaces with general closest point functions},
year = {2012},
fjournal = {SIAM Journal on Numerical Analysis},
journal = {SIAM J. Numer. Anal.},
volume = {50},
number = {6},
pages = {3303--3328},
doi = {10.1137/120865537},
url = {http://people.maths.ox.ac.uk/macdonald/MarzMacdonald-CPfunctions-sinum.pdf},
archiveprefix = {arXiv},
eprint = {1202.3001},
optprimaryclass = {math.NA},
abstract = {The Closest Point Method for solving partial differential
equations (PDEs) posed on surfaces was recently
introduced by Ruuth and Merriman
[J. Comput. Phys. 2008] and successfully applied to
a variety of surface PDEs. In this paper we study
the theoretical foundations of this method. The main
idea is that surface differentials of a surface
function can be replaced with Cartesian
differentials of its closest point extension, i.e.,
its composition with a closest point function. We
introduce a general class of these closest point
functions (a subset of differentiable retractions),
show that these are exactly the functions necessary
to satisfy the above idea, and give a geometric
characterization this class. Finally, we construct
some closest point functions and demonstrate their
effectiveness numerically on surface PDEs}
}
@article{Auer/Macdonald/Treib/Schneider/Westermann:fluidEffects,
author = {Auer, S. and Macdonald, C. B. and Treib, M.
and Schneider, J. and Westermann, R.},
title = {Real-Time Fluid Effects on Surfaces using the {C}losest {P}oint {M}ethod},
year = {2012},
journal = {Comput. Graph. Forum},
fjournal = {Computer Graphics Forum},
volume = {31},
number = {6},
pages = {1909--1923},
doi = {10.1111/j.1467-8659.2012.03071.x},
url = {http://people.maths.ox.ac.uk/macdonald/Auer-2012-fluideffects.pdf},
abstract = {The Closest Point Method (CPM) is a method for
numerically solving partial differential equations
(PDEs) on arbitrary surfaces, independent of the
existence of a surface parametrization. The CPM
uses a closest point representation of the surface,
to solve the unmodified Cartesian version of a
surface PDE in a 3D volume embedding, using simple
and well-understood techniques. In this paper we
present the numerical solution of the wave equation
and the incompressible Navier--Stokes equations on
surfaces via the CPM, and we demonstrate surface
appearance and shape variations in real-time using
this method. To fully exploit the potential of the
CPM, we present a novel GPU realization of the
entire CPM pipeline. We propose a surface-embedding
adaptive 3D spatial grid for efficient
representation of the surface, and present a
high-performance approach using CUDA for converting
surfaces given by triangulations into this
representation. For real-time performance, CUDA is
also used for the numerical procedures of the CPM.
For rendering the surface (and the PDE solution)
directly from the closest point representation
without the need to reconstruct a triangulated
surface, we present a GPU ray-casting method that
works on the adaptive 3D grid.}
}
@article{Macdonald/Brandman/Ruuth:eigen,
author = {Macdonald, Colin B. and Brandman, Jeremy and Ruuth, Steven J.},
title = {Solving eigenvalue problems on curved surfaces using the
{C}losest {P}oint {M}ethod},
journal = {J. Comput. Phys.},
fjournal = {Journal of Computational Physics},
year = {2011},
volume = {230},
number = {22},
pages = {7944--7956},
doi = {10.1016/j.jcp.2011.06.021},
url = {http://people.maths.ox.ac.uk/macdonald/eigensurf.pdf},
archiveprefix = {arXiv},
eprint = {1106.4351},
optprimaryclass = {math.NA},
abstract = {Eigenvalue problems are fundamental to mathematics and
science. We present a simple algorithm for
determining eigenvalues and eigenfunctions of the
Laplace--Beltrami operator on rather general curved
surfaces. Our algorithm, which is based on the
Closest Point Method, relies on an embedding of the
surface in a higher-dimensional space, where
standard Cartesian finite difference and
interpolation schemes can be easily applied. We
show that there is a one-to-one correspondence
between a problem defined in the embedding space and
the original surface problem. For open surfaces, we
present a simple way to impose Dirichlet and Neumann
boundary conditions while maintaining second-order
accuracy. Convergence studies and a series of
examples demonstrate the effectiveness and
generality of our approach.}
}
@article{Ketcheson/Gottlieb/Macdonald:TSRK,
author = {Ketcheson, David I. and Gottlieb, Sigal and Macdonald, Colin B.},
title = {Strong stability preserving two-step {R}unge--{K}utta methods},
fjournal = {SIAM Journal on Numerical Analysis},
journal = {SIAM J. Numer. Anal.},
year = {2012},
volume = {49},
number = {6},
pages = {2618--2639},
doi = {10.1137/10080960X},
url = {http://people.maths.ox.ac.uk/macdonald/KetchesonGottliebMacdonald-2011-tsrk-sinum.pdf},
oldurl = {http://people.maths.ox.ac.uk/macdonald/ssptsrk.pdf},
archiveprefix = {arXiv},
eprint = {1106.3626},
optprimaryclass = {math.NA},
abstract = { We investigate the strong stability preserving (SSP)
property of two-step Runge--Kutta (TSRK) methods.
We prove that SSP TSRK methods belong to a simple
subclass of TSRK methods, in which stages from the
previous step are not used. We derive simple order
conditions for this subclass. Whereas explicit SSP
Runge--Kutta methods have order at most four, we
prove that explicit SSP TSRK methods have order at
most eight. We present TSRK methods of up to eighth
order that were found by numerical search. These
methods have larger SSP coefficients than any known
methods of the same order of accuracy, and may be
implemented in a form with relatively modest storage
requirements. The usefulness of the TSRK methods is
demonstrated through numerical examples, including
integration of very high-order WENO discretizations}
}
@article{Motamed/Macdonald/Ruuth:weno5stability,
author = {Motamed, Mohammad and Macdonald, Colin B. and Ruuth, Steven J.},
title = {On Linear Stability of the Fifth-order {WENO}
Discretization},
journal = {J. Sci. Comput.},
fjournal = {Journal of Scientific Computing},
year = {2010},
volume = {47},
number = {2},
pages = {127--149},
doi = {10.1007/s10915-010-9423-9},
url = {http://people.maths.ox.ac.uk/macdonald/weno5_stability.pdf},
abstract = {We study the linear stability of the fifth-order
Weighted Essentially Non-Oscillatory spatial
discretization (WENO5) combined with explicit time
stepping applied to the one-dimensional advection
equation. We show that it is not necessary for the
stability domain of the time integrator to include a
part of the imaginary axis. In particular, we show
that the combination of WENO5 with either the
forward Euler method or a two-stage, second-order
Runge--Kutta method is linearly stable provided very
small time step-sizes are taken. We also consider
fifth-order multistep time discretizations whose
stability domains do not include the imaginary axis.
These are found to be linearly stable with moderate
time steps when combined with WENO5. In particular,
the fifth-order extrapolated BDF scheme gave
superior results in practice to high-order
Runge--Kutta methods whose stability domain includes
the imaginary axis. Numerical tests are presented
which confirm the analysis.}
}
@article{cbm:ridc,
author = {Christlieb, Andrew J. and Macdonald, Colin B. and Ong, Benjamin W.},
title = {Parallel High-Order Integrators},
journal = {SIAM J. Sci. Comput.},
fjournal = {SIAM Journal on Scientific Computing},
year = {2010},
volume = {32},
number = {2},
pages = {818--835},
doi = {10.1137/09075740X},
url = {http://people.maths.ox.ac.uk/macdonald/ChristliebMacdonaldOng-2010-ridc-sisc.pdf},
oldurl = {http://people.maths.ox.ac.uk/macdonald/ridc.pdf},
abstract = {In this work we discuss a class of defect correction
methods which is easily adapted to create parallel
time integrators for multicore architectures and is
ideally suited for developing methods which can be
order adaptive in time. The method is based on
integral deferred correction (IDC), which was itself
motivated by spectral deferred correction by Dutt,
Greengard, and Rokhlin [BIT, 40 (2000),
pp. 241--266]. The method presented here is a
revised formulation of explicit IDC, dubbed
revisionist IDC (RIDC), which can achieve
$p$th-order accuracy in "wall-clock time" comparable
to a single forward Euler simulation on problems
where the time to evaluate the right-hand side of a
system of differential equations is greater than
latency costs of interprocessor communication, such
as in the case of the $N$-body problem. The key idea
is to rewrite the defect correction framework so
that, after initial start-up costs, each correction
loop can be lagged behind the previous correction
loop in a manner that facilitates running the
predictor and $M=p-1$ correctors in parallel on an
interval which has $K$ steps, where $K$ much greater
than $p$. We prove that given an $r$th-order
Runge--Kutta method in both the prediction and $M$
correction loops of RIDC, then the method is order
$r\times(M+1)$. The parallelization in RIDC uses a
small number of cores (the number of processors used
is limited by the order one wants to achieve). Using
a four-core CPU, it is natural to think about
fourth-order RIDC built with forward Euler, or
eighth-order RIDC constructed with second-order
Runge--Kutta. Numerical tests on an $N$-body
simulation show that RIDC methods can be
significantly faster than popular Runge--Kutta
methods such as the classical fourth-order
Runge--Kutta scheme. In a PDE setting, one can
imagine coupling RIDC time integrators with parallel
spatial evaluators, thereby increasing the
parallelization. The ideas behind RIDC extend to
implicit and semi-implicit IDC and have high
potential in this area.}
}
@inproceedings{luke:segment,
author = {Tian, {Li (Luke)} and Macdonald, Colin B. and Ruuth, Steven J.},
title = {Segmentation on surfaces with the {C}losest {P}oint {M}ethod},
year = {2009},
booktitle = {Proc. ICIP09, 16th {IEEE} International Conference on Image Processing},
pages = {3009--3012},
doi = {10.1109/ICIP.2009.5414447},
address = {Cairo, Egypt},
url = {http://people.maths.ox.ac.uk/macdonald/TianMacdonaldRuuth.pdf},
abstract = {We propose a method to detect objects and patterns in
textures on general surfaces. Our approach applies
the Chan--Vese variational model for active contours
without edges to the problem of segmentation of
scalar surface data. This leads to gradient descent
equations which are level set equations on surfaces.
These equations are evolved using the Closest Point
Method, which is a recent technique for solving
partial differential equations (PDEs) on surfaces.
The final algorithm has a particularly simple form:
it merely alternates a time step of the usual
Chan--Vese model in a small 3D neighborhood of the
surface with an interpolation step. We remark that
the method can treat very general surfaces since it
uses a closest point function to represent the
underlying surface. Various experimental results
are presented, including segmentation on smooth
surfaces, non-smooth surfaces, open surfaces, and
general triangulated surfaces.}
}
@article{cbm:icpm,
author = {Macdonald, Colin B. and Ruuth, Steven J.},
title = {The Implicit {C}losest {P}oint {M}ethod for the Numerical
Solution of Partial Differential Equations on
Surfaces},
journal = {SIAM J. Sci. Comput.},
fjournal = {SIAM Journal on Scientific Computing},
year = {2009},
volume = {31},
number = {6},
pages = {4330--4350},
doi = {10.1137/080740003},
url = {http://people.maths.ox.ac.uk/macdonald/MacdonaldRuuth-2009-icpm-sisc.pdf},
oldurl = {http://people.maths.ox.ac.uk/macdonald/icpm.pdf},
abstract = {Many applications in the natural and applied sciences
require the solutions of partial differential
equations (PDEs) on surfaces or more general
manifolds. The Closest Point Method is a simple and
accurate embedding method for numerically
approximating PDEs on rather general smooth
surfaces. However, the original formulation is
designed to use explicit time stepping. This may
lead to a strict time-step restriction for some
important PDEs such as those involving the
Laplace-Beltrami operator or higher-order derivative
operators. To achieve improved stability and
efficiency, we introduce a new implicit Closest
Point Method for surface PDEs. The method allows
for large, stable time steps while retaining the
principal benefits of the original method. In
particular, it maintains the order of accuracy of
the discretization of the underlying embedding PDE,
it works on sharply defined bands without degrading
the accuracy of the method, and it applies to
general smooth surfaces. It also is very simple and
may be applied to a rather general class of surface
PDEs. Convergence studies for the in-surface heat
equation and a fourth-order biharmonic problem are
given to illustrate the accuracy of the method. We
demonstrate the flexibility and generality of the
method by treating flows involving diffusion,
reaction-diffusion and fourth-order spatial
derivatives on a variety of interesting surfaces
including surfaces of mixed codimension.}
}
@article{Ketcheson/Macdonald/Gottlieb:imssp,
author = {Ketcheson, David I. and Macdonald, Colin B. and Gottlieb, Sigal},
title = {Optimal implicit strong stability preserving {R}unge--{K}utta methods},
journal = {Appl. Numer. Math.},
fjournal = {Applied Numerical Mathematics},
year = {2009},
volume = {59},
number = {2},
pages = {373--392},
doi = {10.1016/j.apnum.2008.03.034},
url = {http://people.maths.ox.ac.uk/macdonald/imssp.pdf},
abstract = {Strong stability preserving (SSP) time discretizations
were developed for use with spatial discretizations
of partial differential equations that are strongly
stable under forward Euler time integration. SSP
methods preserve convex boundedness and
contractivity properties satisfied by forward Euler,
under a modified timestep restriction. We turn to
implicit Runge--Kutta methods to alleviate this
timestep restriction, and present implicit SSP
Runge--Kutta methods which are optimal in the sense
that they preserve convex boundedness properties
under the largest timestep possible among all
methods with a given number of stages and order of
accuracy. We consider methods up to order six (the
maximal order of SSP Runge--Kutta methods) and up to
eleven stages. The numerically optimal methods found
are all diagonally implicit, leading us to
conjecture that optimal implicit SSP Runge--Kutta
methods are diagonally implicit. These methods
allow a significant increase in the SSP timestep
limit, compared to explicit methods of the same
order and number of stages. Numerical tests verify
the order and the SSP property of the methods.}
}
@article{cbm:lscpm,
author = {Macdonald, Colin B. and Ruuth, Steven J.},
title = {Level set equations on surfaces via the {C}losest {P}oint {M}ethod},
journal = {J. Sci. Comput.},
fjournal = {Journal of Scientific Computing},
year = {2008},
volume = {35},
number = {2--3},
pages = {219--240},
doi = {10.1007/s10915-008-9196-6},
url = {http://people.maths.ox.ac.uk/macdonald/lscpm.pdf},
abstract = {Level set methods have been used in a great number of
applications in $\mathbb{R}^2$ and $\mathbb{R}^3$
and it is natural to consider extending some of
these methods to problems defined on surfaces
embedded in $\mathbb{R}^3$ or higher dimensions. In
this paper we consider the treatment of level set
equations on surfaces via a recent technique for
solving partial differential equations (PDEs) on
surfaces, the Closest Point Method
\cite{Ruuth/Merriman:jcp08:CPM}. Our main
modification is to introduce a Weighted Essentially
Non-Oscillatory (WENO) interpolation step into the
Closest Point Method. This, in combination with
standard WENO for Hamilton--Jacobi equations, gives
high-order results (up to fifth-order) on a variety
of smooth test problems including passive transport,
normal flow and redistancing. The algorithms we
propose are straightforward modifications of
standard codes, are carried out in the embedding
space in a well-defined band around the surface and
retain the robustness of the level set method with
respect to the self-intersection of interfaces.
Numerous examples are provided to illustrate the
flexibility of the method with respect to geometry.}
}
@article{cbm:dsrk,
author = {Macdonald, Colin B. and Gottlieb, Sigal and Ruuth, Steven J.},
title = {A numerical study of diagonally split {R}unge--{K}utta methods for {PDEs} with discontinuities},
journal = {J. Sci. Comput.},
fjournal = {Journal of Scientific Computing},
year = {2008},
volume = {36},
number = {1},
pages = {89--112},
doi = {10.1007/s10915-007-9180-6},
url = {http://people.maths.ox.ac.uk/macdonald/dsrk.pdf},
keywords = {diagonally split Runge--Kutta methods; Runge--Kutta
methods; unconditional contractivity; strong
stability preserving; time discretization; order
reduction; stage order; hyperbolic PDEs},
abstract = {Diagonally split Runge--Kutta (DSRK) time discretization
methods are a class of implicit time-stepping
schemes which offer both high-order convergence and
a form of nonlinear stability known as unconditional
contractivity. This combination is not possible
within the classes of Runge--Kutta or linear
multistep methods and therefore appears promising
for the strong stability preserving (SSP)
time-stepping community which is generally concerned
with computing oscillation-free numerical solutions
of PDEs. Using a variety of numerical test
problems, we show that although second- and
third-order unconditionally contractive DSRK methods
do preserve the strong stability property for all
time step-sizes, they suffer from order reduction at
large step-sizes. Indeed, for time-steps larger
than those typically chosen for explicit methods,
these DSRK methods behave like first-order implicit
methods. This is unfortunate, because it is
precisely to allow a large time-step that we choose
to use implicit methods. These results suggest that
unconditionally contractive DSRK methods are limited
in usefulness as they are unable to compete with
either the first-order backward Euler method for
large step-sizes or with Crank--Nicolson or
high-order explicit SSP Runge--Kutta methods for
smaller step-sizes. We also present stage order
conditions for DSRK methods and show that the
observed order reduction is associated with the
necessarily low stage order of the unconditionally
contractive DSRK methods.}
}
@incollection{Macdonald/Spiteri:2001:psrm,
author = {Colin B. Macdonald and Raymond J. Spiteri},
title = {The predicted sequential regularization method for
dif\-fe\-ren\-tial-algebraic equations},
booktitle = {Mathematics and Simulation with Biological,
Economic, and Musicoacoustical Applications},
pages = {107--112},
publisher = {WSES Press},
year = {2001},
opteditor = {C.E. D'Attellis and V.V. Kluev and N. Mastorakis},
editor = {D'Attellis and Kluev and Mastorakis}
}
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